https://web.ma.utexas.edu/mediawiki/index.php?action=history&feed=atom&title=Krylov-Safonov_theorem
Krylov-Safonov theorem - Revision history
2024-03-29T11:19:42Z
Revision history for this page on the wiki
MediaWiki 1.40.1
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1104&oldid=prev
imported>Luis: /* $C^{1,\alpha}$ estimates for fully nonlinear equations */
2012-09-07T20:31:23Z
<p><span dir="auto"><span class="autocomment">$C^{1,\alpha}$ estimates for fully nonlinear equations</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:31, 7 September 2012</td>
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<td colspan="2" class="diff-lineno">Line 66:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vector $e$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vector $e$ <ins style="font-weight: bold; text-decoration: none;">(some previous argument is needed to get that $u$ is Lipchitz)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and</div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1103&oldid=prev
imported>Luis: /* Holder estimates */
2012-09-07T20:30:07Z
<p><span dir="auto"><span class="autocomment">Holder estimates</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:30, 7 September 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l51">Line 51:</td>
<td colspan="2" class="diff-lineno">Line 51:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Then the function $u$ is Holder continuous and satisfies the estimate</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Then the function $u$ is Holder continuous and satisfies the estimate</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^{n+1}(Q_1)}).\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^{n+1}(Q_1)}).\]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, and $n$ (dimension)<del style="font-weight: bold; text-decoration: none;">$</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, and $n$ (dimension).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Harnack inequality ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Harnack inequality ===</div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1102&oldid=prev
imported>Luis: /* Parabolic case */
2012-09-07T20:29:45Z
<p><span dir="auto"><span class="autocomment">Parabolic case</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:29, 7 September 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given a bounded solution of the parabolic PDE</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given a bounded solution of the parabolic PDE</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} <del style="font-weight: bold; text-decoration: none;">u (x,t) + b(x,t) \cdot \nabla </del>u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where repeated indices denotes summation and we assume</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where repeated indices denotes summation and we assume</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">b &\in L^n(Q_1), \\</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>f &\in L^<ins style="font-weight: bold; text-decoration: none;">{</ins>n<ins style="font-weight: bold; text-decoration: none;">+1}</ins>(Q_1).</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>f &\in L^n(Q_1).</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Then the function $u$ is Holder continuous and satisfies the estimate</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Then the function $u$ is Holder continuous and satisfies the estimate</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^n(Q_1)}).\]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^<ins style="font-weight: bold; text-decoration: none;">{</ins>n<ins style="font-weight: bold; text-decoration: none;">+1}</ins>(Q_1)}).\]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) <del style="font-weight: bold; text-decoration: none;">and $||b||_{L^n}</del>$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, <ins style="font-weight: bold; text-decoration: none;">and </ins>$n$ (dimension)$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Harnack inequality ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Harnack inequality ===</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given a nonnegative solution of the parabolic PDE</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Given a nonnegative solution of the parabolic PDE</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} <del style="font-weight: bold; text-decoration: none;">u (x,t) + b(x,t) \cdot \nabla </del>u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[ \sup_{Q_{1/2}(0,-1/2)} u \leq C \left(\inf_{Q_{1/2}(0,0)} u+||f||_{L^n(Q_1)} \right).\]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[ \sup_{Q_{1/2}(0,-1/2)} u \leq C \left(\inf_{Q_{1/2}(0,0)} u+||f||_{L^<ins style="font-weight: bold; text-decoration: none;">{</ins>n<ins style="font-weight: bold; text-decoration: none;">+1}</ins>(Q_1)} \right).\]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$<del style="font-weight: bold; text-decoration: none;">, </del>$n$ (dimension) <del style="font-weight: bold; text-decoration: none;">and $||b||_{L^n}$</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$ <ins style="font-weight: bold; text-decoration: none;">and </ins>$n$ (dimension).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== $C^{1,\alpha}$ estimates for fully nonlinear equations ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== $C^{1,\alpha}$ estimates for fully nonlinear equations ==</div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1101&oldid=prev
imported>Luis at 21:45, 14 March 2012
2012-03-14T21:45:02Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:45, 14 March 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Krylov-Safonov theorem <ins style="font-weight: bold; text-decoration: none;"><ref name="KS"/> <ref name="KS2"/> </ins>provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l72">Line 72:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{reflist|refs=</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="KS">{{Citation | last1=Safonov | first1=M. V. | last2=Krylov | first2=N. V. | title=An estimate for the probability of a diffusion process hitting a set of positive measure | year=1979 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=245 | issue=1 | pages=18–20}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ref name="KS2">{{Citation | last1=Safonov | first1=M. V. | last2=Krylov | first2=N. V. | title=A property of the solutions of parabolic equations with measurable coefficients | year=1980 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=44 | issue=1 | pages=161–175}}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">}}</ins></div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1100&oldid=prev
imported>Luis at 19:50, 14 March 2012
2012-03-14T19:50:05Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:50, 14 March 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Elliptic case ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Elliptic case ==</div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1099&oldid=prev
imported>Luis at 16:51, 14 March 2012
2012-03-14T16:51:41Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:51, 14 March 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Elliptic case ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Elliptic case ==</div></td></tr>
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imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1098&oldid=prev
imported>Luis: /* Harnack inequality */
2012-02-24T16:17:23Z
<p><span dir="auto"><span class="autocomment">Harnack inequality</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:17, 24 February 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[ \sup_{Q_{1/2}(0,<del style="font-weight: bold; text-decoration: none;">0</del>)} u \leq C \left(\inf_{Q_{1/2}(0,<del style="font-weight: bold; text-decoration: none;">-1/2</del>)} u+||f||_{L^n(Q_1)} \right).\]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[ \sup_{Q_{1/2}(0,<ins style="font-weight: bold; text-decoration: none;">-1/2</ins>)} u \leq C \left(\inf_{Q_{1/2}(0,<ins style="font-weight: bold; text-decoration: none;">0</ins>)} u+||f||_{L^n(Q_1)} \right).\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
</table>
imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1097&oldid=prev
imported>Luis: /* $C^{1,\alpha}$ estimates for fully nonlinear equations */
2012-02-23T18:23:51Z
<p><span dir="auto"><span class="autocomment">$C^{1,\alpha}$ estimates for fully nonlinear equations</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:23, 23 February 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l63">Line 63:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any <del style="font-weight: bold; text-decoration: none;">vecor </del>$e$.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any <ins style="font-weight: bold; text-decoration: none;">vector </ins>$e$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.</div></td></tr>
</table>
imported>Luis
https://web.ma.utexas.edu/mediawiki/index.php?title=Krylov-Safonov_theorem&diff=1096&oldid=prev
imported>Luis: Created page with "Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regu..."
2012-02-23T18:23:36Z
<p>Created page with "Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regu..."</p>
<p><b>New page</b></p><div>Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.<br />
<br />
== Elliptic case ==<br />
=== Holder continuity ===<br />
Given a bounded solution of the following elliptic PDE<br />
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]<br />
where repeated indices denotes summation and we assume<br />
\begin{align*}<br />
\lambda I &\leq \{a_{ij}(x)\} \leq \Lambda I \text{ for all $x$. (This is the uniform ellipticity condition)}\\<br />
b &\in L^n(B_1), \\<br />
f &\in L^n(B_1).<br />
\end{align*}<br />
Then the function $u$ is Holder continuous and for some small $\alpha>0$ it satisfies the estimate<br />
\[ ||u||_{C^\alpha(B_{1/2})} \leq C (||u||_{L^\infty(B_1)}+||f||_{L^n(B_1)}).\]<br />
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.<br />
<br />
=== Harnack inequality ===<br />
Given a nonnegative solution of the following elliptic PDE<br />
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]<br />
Under the same assumptions as for the Holder estimates, the following Harnack inequality holds<br />
\[ \sup_{B_{1/2}} u \leq C (\inf_{B_{1/2}} u+||f||_{L^n(B_1)}).\]<br />
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.<br />
<br />
=== Viscosity solutions ===<br />
Both the Holder estimates and the Harnack inequality can be applied to [[viscosity solutions]] of nonlinear equations. Formally, one can replace the equation (at least when $b=0$) by<br />
\begin{align*}<br />
M^+(D^2 u) &\geq f \text{ in } B_1,\\<br />
M^-(D^2 u) &\leq f \text{ in } B_1.<br />
\end{align*}<br />
When $f$ is continuous, both inequalities above are well defined in the viscosity sense.<br />
<br />
<br />
== Parabolic case ==<br />
The elliptic case is implied by the results in the parabolic setting.<br />
<br />
=== Holder estimates ===<br />
The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder<br />
\[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]<br />
<br />
Given a bounded solution of the parabolic PDE<br />
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]<br />
where repeated indices denotes summation and we assume<br />
\begin{align*}<br />
\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\<br />
b &\in L^n(Q_1), \\<br />
f &\in L^n(Q_1).<br />
\end{align*}<br />
Then the function $u$ is Holder continuous and satisfies the estimate<br />
\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^n(Q_1)}).\]<br />
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.<br />
<br />
===Harnack inequality ===<br />
In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.<br />
<br />
Given a nonnegative solution of the parabolic PDE<br />
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]<br />
under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality<br />
\[ \sup_{Q_{1/2}(0,0)} u \leq C \left(\inf_{Q_{1/2}(0,-1/2)} u+||f||_{L^n(Q_1)} \right).\]<br />
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.<br />
<br />
== $C^{1,\alpha}$ estimates for fully nonlinear equations ==<br />
<br />
The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.<br />
\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]<br />
The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vecor $e$.<br />
<br />
Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and<br />
\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]<br />
The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.</div>
imported>Luis